Backgammon-based table game or electronic game

ABSTRACT

A wagering table game based on backgammon is played in which a player wagers on outcomes against a dealer. A backgammon-type board is provided to at least one player that has fewer than 24 position points on the board. Each of the player and dealer are provided with fewer than 15 game pieces. Randomly generated numbers are used to allow players to move player pieces and cause the dealer to move pieces according to fixed rules. A player places a wager against a paytable, odds are paid out on player winning events, and player winning events are determined by removal of all player game pieces from the game board by movement of player game pieces on the game board.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to the field of wagering, casino tablegames and electronic format wagering systems.

2. Background of the Art

Table games, such as Blackjack, Baccarat, Craps, roulette and 3-CardPoker™ games are in widespread use in the gaming industry. The format ofa table game is that there are multiple independent players playingagainst a house or dealer using a fixed strategy. The players havechoices, such as wager placement, amounts of wagers, and whether to hitor stand in Blackjack, but the dealer plays a fixed strategy (e.g., inBlackjack, hitting on 16 and standing on 17 and above; and in Baccarattaking a single card when a hand count is below a minimum count).Another aspect of a table game is that it naturally favors the house. Inother words, when a player makes optimal decisions, the expected returnis only between 95 and 99% of the bet, such that a player may gathershort term gains, but long range advantage always accrues to the house.Finally, Table games must be simple to learn, and are preferably basedon familiar games.

Backgammon is considered one of the oldest games known to mankind. Thegame is in broad use worldwide, including Europe and the Middle East,where it is also known as Tavli or Shesh-Pesh. It is the an aspect ofthis invention to provide a table game based on Backgammon, which meetsthe four criteria of being familiar, provides limited choices to theplayer while the dealer uses a fixed strategy, provides expected returnsin the desired range, and is simple to learn.

SUMMARY OF THE INVENTION

A wagering table game based on backgammon-type game is played on aphysical format, electronic format, or mixed physical and electronicformat game board. A player wagers on outcomes against a dealer. Abackgammon-type board is provided to at least one player that has fewerthan 24 position points on the board. Each of the player and dealer areprovided with fewer than 15 game pieces. Randomly generated numbers areused to allow players to move player pieces and cause the dealer to movepieces according to fixed rules. A player places a wager against apaytable, odds are paid out on player winning events, and player winningevents are determined by removal of all player game pieces from the gameboard by movement of player game pieces on the game board.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows five hand signals that can be used in a Nannon® wageringgame or be icons on electronic game signals indicating the movesdesignated by the hand signals.

FIG. 2 shows a single position game layout on a table position for aNannon® game as would be presented on a casino table felt.

DETAILED DESCRIPTION OF THE INVENTION

The Nannon® game is a simplified version of backgammon which uses afewer number of checkers per side instead of the traditional 15 checkersused in backgammon. The Nannon® game also uses a shorter board of 6-10,instead of 24 spaces. In order to get the blocking and turnabout effectstraditional in backgammon, which requires 12 checkers stacked up on sixpoints, the Nannon® game uses adjacency of single checkers to form ablock. Thus if an opponent has a checker behind two adjacent checkers,the opponent would miss a turn 33% of the time, and behind threeadjacent checkers, an opponent would miss a turn 50% of the time. TheNannon® game further simplifies the underlying Backgammon game by usinga single die instead of two dice, and has no complex rules for bearingoff game pieces. In practice, it only takes only about 2 minutes tolearn to play the Nannon® game, even though more time is likely to beneeded to develop a good strategy.

The play of the Nannon® game proceeds that players take turns rolling adie, and if possible, choosing a checker to advance the amount shown onthe die. The player may have a choice of moves, which require astrategic decision, or the move may be forced (only a single move ispossible), or there may be no legal move so the player misses a turn.The goal of the Nannon® game is to get checkers off the opposite side ofthe board. Once a single checker is left, the moves are dictated by thedice rolls, and it becomes a game of probabilities and pure luck.

There are unique configurations and specifically definable events in theplay of our Nannon® game, especially where one player has two checkersplaying against an opponent with one checker. The one checker opponentis at an advantage towards winning, but every move is dictated by thedie. The two checker player usually gets to make a few strategicdecisions before the game reverts to pure luck. There are 198 positionspossible, of which 111 positions have decisions to be made by the twochecker player on at least one dice roll. We believe this strategicknowledge to be of the same scale as blackjack or video poker.

This configuration of two checkers against one checker meets the generalcriteria of providing a player with strategic choices while having adealer use a fixed strategy—in this case moves forced by the die roll.Thus there are 3 of the 4 criteria, and are left to find configurationsof this game which will provide the proper return. By default andsimilarity to Blackjack, it is assumed the player moves first.

Calculating the Returns.

It is known to people familiar with the field of Machine Learning thatBackgammon games are Markov games, and in 1957 Bellman proved theexistence of an optimal value array for such Markov games, together withan iterative algorithm which could converge to the values (R. Bellman.Dynamic Programming. Princeton University Press, Princeton, N.J., 1957).This “value iteration” algorithm starts by labeling the endgamepositions with the payoffs and then updating the values for allnon-ending positions using a single ply look-ahead now known as“expectimax.” The result of this iterative process is an array ofexpected payoff values for every possible position in the game, assumingboth players make optimal moves going forward. This value array providesa “greedy” optimal move strategy by supplying the information to choosethe move among alternatives which has maximum payoff.

There are 3 variables which are involved in calculating returns. Thefirst variable is the starting position of the 3 checkers. On a sixpoint board, these are indicated by numbers between 0 and 7. We appliedthis technique to a starting position of 1 & 2 versus 6, and found thatthe dealer would win 80% of the time, which does not provide a playerwith fair odds or much fun. However, while observing optimal games, wediscovered two important things about the two-versus-one game. Firstthat when the player wins, the dealer's checker is always on one of the6 board positions, providing the player with a “lead” of 1 through 6,and second that the percentage of games where the dealer ended on the 1,2, 3 . . . 6 point was monotonically shrinking. This is shown in Table 1for the 1 & 2 versus 6 game:

Lead Likelihood Win by 1 0.0592 Win by 2 0.0448 Win by 3 0.039 Win by 40.0298 Win by 5 0.0228 Win by 6 0.0078 % lose 80% % win 20%We realized that by increasing the payout as the lead was greater, wecould solve the problem of providing a reasonable return. For example,in the 1-2 versus 6 game, if the payout was {2,3,4,5,6,7}, the playercould win up to 7× his bet, but this would be very rare, less than 1% ofthe time. Table 2 shows that calculating the return using odds in thepayout raises the return from 20% to 95.4% which is in the desiredrange.

Lead Likelihood return Equity Win by 1 0.0592 3 0.1776 Win by 2 0.0448 40.1792 Win by 3 0.039 5 0.195 Win by 4 0.0298 6 0.1788 Win by 5 0.0228 70.1596 Win by 6 0.0078 8 0.0624 % lose 80% 0 0 % win 20% return: 95.26%The second variable is the payout table, and many tables are possible,such as {1,2,3,4,5,6}, {1,1,2,2,3,3} or {1,1,2,3,5,8}. However, witheach different payout table, the optimal strategy will change, so we hadto derive the value array anew for each payout table. Furthermore, oncethe values were computed, the starting position would determine theplayer return. After calculating the value array, we searched forstarting positions which provided a return of between 90% and 102%.

There were only a few such positions. One position stood out for thenatural {1,2,3,4,5,6} payout, namely that a starting position of 1-3versus 6 yielded a return of 98.6%, and 27% wins, and we chose this asour de facto standard for the backgammon table game, however, ourinvention is not limited to a six point board or this one startingposition.

In order to generate more starting positions to provide flexibility to ahouse, we added a third variable—limiting the dealer's first roll to 1-5or 1-4 by modifying the expectimax algorithm. We found a good number ofpayouts, starting conditions, and first roll limits which provide ourinvention with the fourth criteria, of providing a return which givesthe house an edge. Forty of these are shown in table 3, sorted by returnfrom 90% to 102%. (In order to generate interest in the game, the casinomight use a >100% payoff for an introductory period.)

Player Dealer Payoff Table Start Dealer First Roll Return 1 1 1 2 2 5 13 6 1-4 90.8% 1 1 2 2 2 25 1 3 5 1-4 91.3% 2 3 4 5 6 7 0 1 7 1-4 91.5% 11 1 2 2 4 1 2 7 1-4 92.0% 1 2 4 8 16 32 0 2 6 1-6 92.0% 1 2 4 8 16 32 12 5 1-6 92.2% 2 3 4 5 6 7 0 2 6 1-4 93.1% 1 1 2 2 2 25 0 3 7 1-6 93.6% 12 3 4 5 6 2 3 5 1-5 93.7% 1 1 2 2 2 25 0 2 7 1-4 94.2% 1 1 2 2 2 25 2 36 1-6 94.2% 0 1 2 3 4 5 1 2 7 1-5 94.4% 1 2 3 4 5 6 0 2 7 1-4 94.7% 1 13 3 5 5 1 2 6 1-4 94.9% 1 1 1 2 2 5 1 2 7 1-4 95.3% 2 3 4 5 6 7 1 2 61-6 95.4% 1 1 2 2 2 25 1 2 6 1-4 95.4% 1 2 3 5 10 25 2 3 5 1-6 95.6% 1 24 8 16 32 0 3 5 1-4 95.9% 1 1 1 2 2 4 1 3 7 1-6 96.3% 1 1 3 3 5 5 1 2 71-6 96.6% 1 1 2 3 5 8 2 3 5 1-4 96.7% 1 2 3 5 10 25 0 1 7 1-4 97.3% 1 23 4 5 6 1 3 6 1-6 98.6% 1 2 3 5 10 25 0 3 6 1-5 98.7% 1 1 2 3 5 8 1 2 71-6 98.8% 1 1 3 3 5 5 2 3 5 1-4 98.9% 1 1 3 3 5 5 0 3 7 1-4 99.6% 1 1 23 5 8 0 3 7 1-4 99.8% 1 2 3 4 5 6 1 3 5 1-4 99.8% 1 1 1 2 2 5 1 3 7 1-6100.2% 1 1 2 2 2 25 2 3 5 1-4 100.5% 1 1 2 3 5 8 2 3 6 1-6 100.6% 1 1 12 2 4 2 3 6 1-4 100.6% 1 2 3 5 10 25 0 2 6 1-4 100.7% 0 1 2 3 4 5 2 3 61-5 100.8% 1 2 3 4 5 6 0 3 7 1-5 101.2% 2 3 4 5 6 7 2 3 5 1-6 101.6% 2 34 5 6 7 0 2 7 1-6 101.7% 1 1 1 2 2 5 2 3 6 1-4 101.7% 1 1 2 2 2 25 1 3 61-5 101.9%

Security Concerns

There are 2 major security concerns in a table game setup where a dealerplays against several players.

First is the fairness of the dice. Dice can be loaded, exchanged,damaged, which affect the outcome. Instead of dice, the table game coulduse 7 decks of custom playing cards with dice pictures, automaticallyshuffled and dealt as needed from a shoe as in other table games.Alternatively an electronic random element (RANDOM NUMBER GENERATOR)could be used, or even a dice Popper, as in the Trouble® game.

Secondly, if the players can move their hands across the table, theymight change a bet in the middle of a game, or move a piece into abetter position. Thus the dealer has to be the only person allowed totouch the checkers and the dice. Electronic wagering systems that areavailable on casino table games are useful in avoiding that issue.

To ameliorate this security concern, the players keep their hands offthe table, and communicate their decisions with words and/or handgestures. There are 5 such decisions which can be kept unambiguous withhand signals similar to hit and stay in Blackjack. These are FRONT,BACK, HIT, BLOCK, and RUN.

FRONT and BACK directly indicate which checker to move, the forwardmost(FRONT) or rearmost (BACK) checker. HIT indicates that one of theplayer's checkers can land on the dealer's checker, sending it off theboard or to a restart position. RUN means to bear off with a checker,and BLOCK means to move the rear checker adjacent to the front checkerinto a block formation, potentially causing the dealer a missed turn.Use of these signals is not essential, except for indicating whichchecker to move. For example, by moving the BACK checker one space, aBLOCK may be automatically formed, without making a signal that a BLOCKwas intended.

FIG. 1 shows hand signals for these 5 decisions.

FIG. 2 shows a layout of a table cloth for the game against 4 players.Each position has a board and a betting area.

A general and detailed description of play according to the presenttechnology can include a method of playing a wagering table game basedon backgammon in which a player wagers on outcomes against a dealer. Abackgammon-type board with points or positions is provided to at leastone player that has fewer than 24 position points on the board. Each ofthe player and dealer are provided with fewer than 15 game pieces.Randomly generated numbers are used to allow players to move playerpieces and cause the dealer to move pieces according to fixed rules,very similar to those of backgammon, with the differences detailed aboveand below. A player places a wager against a paytable, odds are paid outon player winning events, and player winning events are determined byremoval of all player game pieces from the game board by movement ofplayer game pieces on the game board. The game board preferably hasfewer than 10 position points on the game board and each of the playerand dealer have fewer than 4 game pieces, for example, the game boardhas 6 position points on the game board and each of the player anddealer have fewer than 3 game pieces and payout odds are determined bydealer game piece positions at a time when a player removes all playergame pieces from the game board or based on the dealer's game piece(s)initial starting position relative to the player's game pieces startingposition. The preferred wagering game method is where the player has twogame pieces and the dealer has a single game piece. The two playerpieces may begin the game at one end of the game board and the singledealer piece begins the game at another end of the game board. Thepayout odds may be determined by dealer game piece positions at a timewhen a player removes all player game pieces from the game board. Thewagering game may be played wherein the player's game pieces and dealergame piece are initially positioned on the game board by a randomassignment of game pieces within a limited range of available game piecepositions for player game pieces and the dealer game piece. The wageringgame method may be played wherein the player's game pieces and dealergame piece are initially positioned on the game board at positions otherthan only extreme end positions on the game board. The wagering gamemethod may be played wherein a dealer's first available random numberused in play of the game is limited in size to a range of numbers lessthan a number that can move the dealer's single game piece off the gameboard.

Another way of describing a method of playing a wagering game on agaming table or on an electronic gaming system can be presented asfollows:

-   -   a) providing a game board to a player, the game board having        fewer than 10 available positions on the game board for moving        game pieces;    -   b) the player placing a wager against a paytable;    -   c) the player and a dealer receiving fewer than four game pieces        each on the game board;    -   d) the player and the dealer alternatively receiving random        numbers that are used to move respective game pieces from a        starting position on the game board towards removal of game        pieces from the game board;    -   e) receiving random numbers until a first of the player and the        dealer have removed all of their respective game pieces from the        game board; and    -   f) resolving the player's wager against the paytable.        The method game board may be a physical game board or a virtual        on a casino gaming table wherein:    -   the player is given two game pieces;    -   the dealer is given a single game piece;    -   the player game pieces are placed within a first three points on        a game board having six points thereon;    -   the dealer's game piece is placed within a set of last three        points on the game board having six points thereon;    -   random numbers are provided by a single die or a single value        provided by a random number generator; and    -   the player's wager is resolved as a win when all of the player's        game pieces are removed from the game board before the dealer's        game piece is removed from the game board.

The method can be played, for example, wherein a dealer's firstavailable random number used in play of the game is limited in size to arange of numbers less than a number that can move the dealer's singlegame piece off the game board. Adjacent game pieces may form blockadesagainst movement of game pieces. Initial positioning of the player's anddealer's game pieces at the start of the game may determine payoutawards on the paytable for a player winning outcome. The dealer's gamepiece position when all of a player's game pieces have been removed fromthe game board may determine payout awards on the paytable for aplayer's winning outcome. The game board may be an electronic game boardwith a processor and video display, random numbers are generated by arandom number generator in the processor, and wagers are placedelectronically and received by the processor. The method may bestrategically played wherein the player's two game pieces are positionat points 1 and 3 on the game board and the dealer's game piece ispositioned at point 6 on the game board.

In an electronic gaming system, many different combinations oralternatives in construction may be played. Player input can be throughbuttons, touchscreens, personal data entry systems (PDA's, cell phones,etc.), player terminals on a communal system, stand-alone units, and thelike. Gaming systems will have at least one processor and informationdisplay to provide information to at least the player on wagers,outcomes and game progression. Rules will be controlled or executed(e.g., dealer's will have all moves controlled by the rules withoutexercise of judgment) and accounting of wagers and outcomes will bedetermined by a processor.

1. A method of playing a wagering table game based on backgammon inwhich a player wagers on outcomes against a dealer, wherein: abackgammon board is provided to at least one player that has fewer than24 position points on the board; each of the player and dealer areprovided with fewer than 15 game pieces; randomly generated numbers areused to allow players to move player pieces and cause the dealer to movepieces according to fixed rules; wherein a player places a wager againsta paytable, odds are paid out on player winning events, and playerwinning events are determined by removal of all player game pieces fromthe game board by movement of player game pieces on the game board. 2.The wagering game method of claim 1 wherein the game board has fewerthan 10 position points on the game board and each of the player anddealer have fewer than 4 game pieces.
 3. The wagering game method ofclaim 1 wherein the game board has 6 position points on the game boardand each of the player and dealer have fewer than 3 game pieces andpayout odds are determined by dealer game piece positions at a time whena player removes all player game pieces from the game board.
 4. Thewagering game method of claim 3 wherein the player has two game piecesand the dealer has a single game piece.
 5. The wagering game method ofclaim 4 wherein the two player pieces begin the game at one end of thegame board and the single dealer piece begins the game at another end ofthe game board.
 6. The wagering game method of claim 3 wherein payoutodds are determined by dealer game piece positions at a time when aplayer removes all player game pieces from the game board.
 7. Thewagering game method of claim 4 wherein payout odds are determined bydealer game piece positions at a time when a player removes all playergame pieces from the game board.
 8. The wagering game method of claim 5wherein payout odds are determined by dealer game piece positions at atime when a player removes all player game pieces from the game board.9. The wagering game method of claim 6 wherein the player's game piecesand dealer game piece are initially positioned on the game board by arandom assignment of game pieces within a limited range of availablegame piece positions for player game pieces and the dealer game piece.10. The wagering game method of claim 6 wherein the player's game piecesand dealer game piece are initially positioned on the game board atpositions other than only extreme end positions on the game board. 11.The wagering game method of claim 4 wherein a dealer's first availablerandom number used in play of the game is Limited in size to a range ofnumbers less than a number that can move the dealer's single game pieceoff the game board.
 12. The wagering game method of claim 7 wherein adealer's first available random number used in play of the game islimited in size to a range of numbers less than a number that can movethe dealer's single game piece off the game board.
 13. A method ofplaying a wagering game on a gaming table or on an electronic gamingsystem comprising: a) providing a game board to a player, the game boardhaving fewer than 10 available positions on the game board for movinggame pieces; b) the player placing a wager against a paytable; c) theplayer and a dealer receiving fewer than four game pieces each on thegame board; d) the player and the dealer alternatively receiving randomnumbers that are used to move respective game pieces from a startingposition on the game board towards removal of game pieces from the gameboard; e) receiving random numbers until a first of the player and thedealer have removed all of their respective game pieces from the gameboard; and f) resolving the player's wager against the paytable.
 14. Themethod of claim 13 wherein the game board is a physical game board on acasino gaming table; the player is given two game pieces; the dealer isgiven a single game piece; the player game pieces are placed within afirst three points on a game board having six points thereon; thedealer's game piece is placed within a set of last three points on thegame board having six points thereon; random numbers are provided by asingle die; and the player's wager is resolved as a win when all of theplayer's game pieces are removed from the game board before the dealer'sgame piece is removed from the game board.
 15. The method of claim 14wherein a dealer's first available random number used in play of thegame is limited in size to a range of numbers less than a number thatcan move the dealer's single game piece off the game board.
 16. Themethod of claim 14 wherein adjacent game pieces form blockades againstmovement of game pieces.
 17. The method of claim 14 wherein initialpositioning of the dealer's game piece at the start of the gamedetermines payout awards on the paytable for a player winning outcome.18. The method of claim 14 wherein the dealer's game piece position whenall players' game pieces have been removed from the game boarddetermines payout awards on the paytable for a player's winning outcome.19. The method of claim 14 wherein the game board is an electronic gameboard with a processor and video display, random numbers are generatedby a random number generator in the processor, and wagers are placedelectronically and received by the processor.
 20. The method of claim 14wherein the player's two game pieces are position at points 1 and 3 onthe game board and the dealer's game piece is positioned at point 6 onthe game board.